Describe in detail the difference between electric potential and potential energy by giving examples and presenting relevant formulae.

Potential energy and potential difference have this relationship: U = qV, where U is potential energy, q is charge and V is the potential difference. Potential energy is the same thing we've seen in mechanics, it is a type of currency with which to do things, it can transfer into kinetic energy when it sets something into motion, it can be used to push objects and do work, etc. Electric potential is useful theoretical quantity to measure in the influence of the electric field (which can directly produce force via F = qE) between relative areas. V is the negative integral of E, and when we sum up all those pieces of E, we can find the "difference" in its strength between points, and therefore we can tell that a charge would flow from an area of high potential (or higher electric field) to areas of lower potential (the convention of going from high to low is where the minus sign on the integral comes from, and of course reverses if the charge is negative), like how water flows from areas of high pressure to low pressure. I like to read electric potential as "this is how much electric pressure is here" and that pressure can indirectly set charges into motion, but E is the actual thing that generates that force. V is a lot like an energy map, it uses scalar values (not vectors) to describe the influence of the electric field  and is often the most useful description of physical situations in electrodynamics (no vectors means we can side step a lot of complex vector calculus). It sits as the middle man between electric field, which allows us to use vector quantities like force and acceleration to describe things, and potential energy which is the gateway to all the different energy descriptions we have like work, kinetic energy and such. From a solid description of V, we can find almost anything about an electric dynamic system with relative ease.

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